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\title{《基础复分析》第9章共形映射与Dirichlet问题 - 习题}
\author{CGZ ET AL}

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%## 《基础复分析》习题九

\begin{enumerate}

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\item % 1

用 Schwarz-Christoffel 公式导出将单位圆盘映为水平带域以及其具有正实部的半带域的共形映射。
    

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\item % 2

用 Schwarz-Christoffel 公式导出将单位圆盘映为扩充复平面除去一条线段或者射线的共形映射。
    

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\item % 3

证明
$$
F(w) = \int_0^w (1 - w^n)^{-\frac{2}{n}} \, dw
$$
将单位圆盘映为正 $n$ 边形的内部。
    

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\item % 4

试求将上半平面映为区域
$$
\Omega = \{z = x + iy : x > 0, y > 0, \min\{x, y\} < 1\}
$$
的共形映射。
    

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\item % 5

设 $E$ 是区域 $\Omega$ 内的紧集. 证明存在只依赖于 $E$ 及 $\Omega$ 的常数 $M < \infty$, 使得对 $\Omega$ 内任意非负调和函数 $u(z)$, 以及任意两点 $z_1, z_2 \in E$, 有 $u(z_2) \leq M u(z_1)$.
    

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\item % 6

证明函数 $|x|$, $|z|^{\alpha}$ ($\alpha > 0$), $\ln(1 + |z|^2)$ 都是次调和的。
    

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\item % 7

如果 $f(z)$ 是解析函数, 证明 $|f(z)|^{\alpha}$ ($\alpha > 0$) 及 $\ln(1 + |f(z)|^2)$ 都是次调和的。
    

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\item % 8

如果 $v(z)$ 具有二阶连续偏导数, 证明 $v(z)$ 是次调和的当且仅当 $\Delta v \geq 0$.
    

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\item % 9

证明次调和函数在自变量的共形映射变换下仍然是次调和的。
    

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\item % 10

证明两个圆环共形等价, 当且仅当它们的半径之比相等。




\end{enumerate}

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